A gymnasium in the shape of a rectangular parallelepiped with volume 960,000 cub

Question

A gymnasium in the shape of a rectangular parallelepiped with volume 960,000 cubic feet is to be erected. Assume that because of decorations, the front wall will cost twice as much per square foot as the side and back walls, and the floor and the roof will cost 1.5 times as much as the side walls. Find the dimensions of the gymnasium that will minimize the cost. a) Do this problem using the second derivatives test. b) Do this problem using Lagrange multipliers. (Be sure to communicate: what variables represent, what you are doing to equations in your model (Label them), write an English sentence concluding your result as well as its reasonableness.)

Explanation / Answer

cost of front part and roof is   L*H + L * W = 2( 2HW+LH)

LWH = 960,000

2*2HW = 2LW we have these three equation.

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